Regression Uncertainty Estimation

Gaussian processes on graphs provide a principled framework for prediction and uncertainty estimation, but modelling heterogeneous networks often requires balancing flexibility against parameter complexity. Variability at a node may reflect useful signal that should propagate through the network, unreliable measurements whose influence should be suppressed, or a mixture of both. We introduce Variance-Coupled Gaussian Processes (VCGPs), which use a positive auxiliary signal to modulate graph interactions through a single learned coupling exponent. The exponent controls how strongly high-variability nodes propagate or attenuate information, while a cluster-wise extension accommodates different regimes within the same graph. Under stated structural assumptions, we characterize the unique stationary-rescaling point of the basic family, where correlations match those of a stationary graph kernel while node-dependent variances remain. Experiments on traffic, air-quality and weather datasets show competitive prediction and uncertainty estimation relative to stationary and per-node non-stationary baselines, with the largest gains on networks requiring multiple coupling regimes. Stationary correlations after variance rescaling remain sufficient on some data, and performance depends on the auxiliary signal and model selection.

Machine-learning surrogates of linear peeling-ballooning stability now run in milliseconds, but they are mostly deployed as point estimators: they do not say how wrong they may be, or when a query lies outside what they were trained on. For a fusion company designing a new device, every equilibrium of interest is outside the training data of existing surrogates, so this is the case that matters. We describe a model-agnostic trust layer for stability surrogates, made of conformal error bars, out-of-distribution alarms that route doubtful cases to the physics code, and recalibration from a few simulations of a new design, and we test it with pre-registered hypotheses. On a large public dataset of JET-like equilibria with ideal-MHD growth rates, the error bars hold their nominal coverage in distribution and fall short under a shift in plasma shape; a few dozen simulations of the new shape, or routing the cases the alarms flag, restore it. On resistive BOUT++ surrogates, the bars stay calibrated on held-out runs, the alarm flags tokamaks absent from training, and a scenario scan is screened in seconds instead of hours. Across equilibria from the SPARC library and the DIII-D coil set, a surrogate trained on our own graded BOUT++ labels beats the baseline on unseen equilibria, while a surrogate trained elsewhere ranks the same cases backwards and its alarm declines them. Building our own neural-operator surrogate, the harness exposed two input defects whose fix closed its gap to a simple network. For every surrogate tested, the trust layer keeps coverage near target, with a bar width set by that surrogate's own error. Not every pre-registered hypothesis held, and we report those results and the limits of all of them.

Uncertainty estimates from fuzzy regression are informative only when the implementation is correct and the comparison protocol makes the sources of variation visible. We revisit a first-order Takagi--Sugeno--Kang (TSK) regression implementation and rebuild its dense, conjugate Bayesian, BIC-based, and rule-level spike-and-slab estimators using corrected code. The corrected pipeline freezes the Gaussian membership partition at training time, uses a scaled conjugate prior, evaluates firing strengths in the log domain, and reports direct posterior-predictive quantiles for Bayesian methods. Across 30 outer 80/20 splits on three targets from two UCI regression datasets, corrected TSK-LS obtains $R^2=0.957$, $0.923$, and $0.786$ on Energy-Heating, Energy-Cooling, and Concrete, respectively; rule-level Gibbs sampling with Bayesian model averaging obtains $0.957$, $0.922$, and $0.784$. We then evaluate the same workflow on a bounded chronological sample of the Building Data Genome 2 meter data that covers 158 buildings across 17 sites. In a common-unit evaluation, every method is scored on the same 47,400 test rows and every trained model uses 94,800 train--validation rows. Persistence has RMSE $27.710$, compared with $29.083$ for both TSK-LS and Bayesian TSK; the pre-specified building-level mean RMSE difference between Bayesian TSK and persistence is $+0.0447$ with a 95% site-cluster bootstrap interval of $[-1.5536,1.5665]$. The Bayesian TSK 90% intervals have PICP $0.980$ and MPIW $116.469$. Taken together, the evidence supports a narrower conclusion: spike-and-slab model averaging is a reproducible implementation option, but it does not provide a demonstrated accuracy or calibration advantage across the tested tabular and building-meter settings.

Reliable speed-over-ground estimation from ground-facing Doppler spectra is essential for mobile platforms such as railway systems. Classical signal-processing approaches can provide accurate estimates under favorable conditions, but often degrade in challenging conditions. We propose confidence-aware speed-over-ground estimation trained on rail-domain data that estimates speed over ground and frame-level confidence from Doppler spectra with several novel contributions: First, we transfer the idea of exploiting a reconstruction task by an autoencoder for performance estimation from previous work to Doppler RaDAR. Second, we append a speed-over-ground estimation head operating on the autoencoder's latent space features. Third, we omit the reconstruction decoder during inference but add a low-cost dedicated confidence head for frame-wise error-bound estimation. Fourth, we reveal how noise augmentation training can provide high robustness to the method. Experiments show that our approach improves speed estimation accuracy over classical baselines while providing informative confidence estimates that indicate the reliability of individual predictions, even under adverse signal conditions.

An uncertainty estimate is meaningful only if its semantics can be tied to observable predictive behavior. Normal-Inverse-Gamma (NIG) evidential regression provides a structured decomposition of predictive variability into data-scale and predictive-mean components, but marginal likelihood does not identify their allocation. We address this semantic ambiguity in masked representation learning. MORSE (Masked Observability for Reliable Structured Encoding) predicts the same hidden target under controlled changes in visible context and uses normalized cross-context variation to ground predictive-mean uncertainty, while a Gaussian objective learns the conditional mean and data scale. The normalized variation plays the role of the NIG factor $1/ν$, which the marginal leaves unidentified; this yields a policy-relative observability variance that measures dependence on visible evidence, distinct from variation induced by resampling the training set. On a Gaussian-process benchmark with a contextual reference built from analytic Bayes conditional means, MORSE has the lowest log-variance error among compared evidential, ensemble, and dropout estimators in all nine tested settings, while the estimator closest to training-set variation perturbs the model rather than its context in every setting, so the two references separate the estimators empirically. MORSE amortizes observability estimation into one forward pass per observation state. On a heteroscedastic pulse benchmark, its predictor's cross-context variance approaches a large-sample reference as training data increase. MORSE also improves low-label transfer on PTB-XL and in-distribution trajectory prediction on IMPTC.

Nonlinear dynamical systems with memory arise across science and engineering, yet uncertainty quantification for efficient forecasting methods such as Next Generation Reservoir Computing (NGRC) remains underdeveloped. We study Bayesian ridge and conformal prediction intervals for NGRC and characterize when their uncertainty estimates agree or differ. In low dimensions, their asymptotic widths are governed by different summaries of the residual distribution, so agreement depends on residual shape rather than dimensionality alone. In high dimensions, regularization introduces a further tradeoff between estimation variance, shrinkage bias, and posterior uncertainty, leading to an explicit transition between regimes where Bayesian intervals are wider or narrower than conformal intervals. We extend these results to quadratic NGRC feature maps and give sufficient conditions for transferring the analysis to temporally dependent forecast windows. Simulations and real-data experiments support the theoretical predictions and illustrate how residual distribution, regularization, dimensionality, and distribution shift affect interval calibration and efficiency. These results provide a principled framework for choosing and interpreting uncertainty quantification methods in reservoir-based forecasting.

Accurate estimation of left ventricular ejection fraction (EF) from echocardiography is central to cardiovascular care, and deep learning enables automated EF prediction from echocardiographic video. Because EF is clinically derived from left-ventricular (LV) volumes, a widely held intuition is that explicit LV segmentation should improve prediction. We introduce a quantitative criterion, the segmentation ceiling, that makes this testable: from EF as a normalized difference of end-diastolic and end-systolic volumes, we derive in closed form how per-frame segmentation area error propagates into EF error, and thus the accuracy a mask must reach before it can improve on direct regression. Using EchoNet-Dynamic, a UniFormer-S backbone, and the empirically measured within-patient error correlation, the criterion places the break-even near 10% per-frame area error, whereas a representative segmenter operates at roughly 14%, above the ceiling. Consistent with this, four strategies for injecting segmentation or area information (a predicted-mask channel, end-diastolic/end-systolic clip sampling, and per-bin and amplitude area-consistency objectives) fail to beat a raw-video baseline; ground-truth masks help only through label leakage. Input representation thus not being the limit, we identify generalization as the practical lever: weight averaging with strong augmentation attains a test R^2 of 0.806 (MAE 4.08) under a matched dense-clip protocol, comparable to an R(2+1)D baseline (0.811) while tightening the validation-to-test gap. Finally, a heteroscedastic beta-NLL formulation yields informative, well-calibrated per-prediction uncertainty, larger for clinically harder low-EF cases, where Monte-Carlo dropout does not. The segmentation ceiling gives a concrete design criterion for when mask-guided EF estimation is worthwhile, plus a simple, uncertainty-aware recipe for EF regression.

Reliable uncertainty estimates are critical in safety-sensitive applications. For such estimates to be useful in practice, it is crucial to understand the sources underlying a model's uncertainty, motivating the disentanglement of total uncertainty into epistemic and aleatoric uncertainty. Existing notions of uncertainty differ in the sources they capture and, consequently, in their definitions of aleatoric and epistemic uncertainty, with no universally accepted definition. We define uncertainty through sample-conditional pointwise posterior risk, which is the expected loss of a predictor under the distribution of plausible ground-truth functions given the observed sample. This definition unifies probabilistic and risk-based concepts of uncertainty. To assess state-of-the-art uncertainty disentanglement methods, we develop a framework that directly compares their estimates against ground-truth uncertainty defined primarily by posterior risk, alongside commonly used alternative uncertainty definitions. We find that Spectral-normalized Neural Gaussian Processes and Variational Latent Gaussian Processes most closely recover the ground-truth uncertainty, while most methods track posterior variance more closely than posterior risk, missing the predictors' bias. Beyond method rankings, we investigate how strongly estimated aleatoric and epistemic uncertainty are entangled and how sensitive uncertainty quality is to modeling choices, yielding practical guidance for uncertainty disentanglement. To support further method development and validation, we release 13 semi-synthetic UCI/OpenML datasets with known posteriors, enabling the computation of ground-truth uncertainty. Code and data will be made publicly available upon acceptance.

Accurate estimation of forest height from satellite imagery is essential for applications such as carbon accounting, biodiversity monitoring, and ecosystem management. While recent deep learning approaches provide accurate predictions, they typically do not quantify predictive uncertainty. This limitation is particularly relevant in geospatial settings characterized by sparse supervision and geographic distribution shift. In this work, we investigate Deep Evidential Regression (DER) for forest height estimation on the TreeUQ benchmark, a large-scale dataset designed for the joint estimation of tree count and average tree height at 10 m resolution, based on Sentinel-1/-2 data as well as tree inventory data over the federal state of Bavaria. To account for the extreme label sparsity of the tree inventory data, we introduce a masked evidential loss for dense geospatial prediction. Using a U-Net architecture with multimodal Sentinel-1 and Sentinel-2 inputs, the proposed approach jointly predicts tree height and associated uncertainty estimates in a single forward pass. Experimental results show that DER achieves predictive performance comparable to a deterministic U-Net while additionally providing well-calibrated uncertainty estimates. These findings demonstrate the potential of evidential learning as an efficient framework for uncertainty-aware forest structure estimation from Earth observation data.

Earth Observation regression tasks such as building height, canopy height, and above-ground biomass estimation underpin critical applications in urban planning, forest monitoring, and climate policy, where both accuracy and reliability are critical. Yet most deep learning models yield only deterministic predictions, providing no indication of per-pixel reliability. These regression tasks are inherently challenging due to heterogeneous land surfaces, skewed target distributions, sensor noise, and signal saturation at high target values, making uncertainty (UC) estimation essential for reliable inference. We address this gap by modeling aleatoric uncertainty using year-long Sentinel-1 SAR and Sentinel-2 MSI time series, proposing two complementary approaches: (i) Gaussian UC, which jointly predicts mean and standard deviation under a Gaussian assumption, and (ii) Quantile UC, which estimates the 10th, 50th, and 90th quantiles to capture asymmetric and heteroscedastic error distributions. Both models are evaluated on three representative EO regression tasks at 10 m spatial resolution. Results show that both approaches match or surpass deterministic benchmarks and existing global products, while delivering well-calibrated, interpretable, and operationally useful confidence estimates. Notably, both models outperform the current 10 m state-of-the-art uncertainty-aware model for canopy height estimation. Our implementation will be available at: https://github.com/RituYadav92/EO-Regression-Uncertainty-Estimation

Efficient uncertainty quantification (UQ) is essential for trustworthy large-scale learning. Existing UQ methods for regression tasks mainly operate under the assumption that the conditional label marginal satisfies single-peak parametric models, e.g., Gaussians, where the negative log-likelihood function simplifies to the mean square error. However, such single-peak assumptions fail in regression tasks featuring multi-modal distributions. On the other hand, semi-parametric methods which achieve strong regression performance for multi-modal distributions often lack efficient quantification on their prediction variances. In this work, we extend UQ techniques based on Variational Bayesian Inference (VBI) to two widely used semi-parametric regression models that yield histogram-like reconstructions of the conditional label densities: Quantile Regression (QR) and Classification Restoration (CR). Our approach introduces a unified, distribution-agnostic framework that simultaneously achieves accurate estimation of complex conditional distributions and highly efficient UQ. Theoretically, our method is grounded in novel formulations of QR and CR within the VBI framework, yielding analytic Evidence Lower Bounds (ELBO) to streamline training and a closed-form or analytically approximated predictive density for efficient inference. Empirically, we evaluate our methods on three large-scale regression benchmarks with multi-modal label distributions. Our framework outperforms state-of-the-art multi-modal regression baselines, and even matches predictive performance of computationally expensive ensemble models. Furthermore, by leveraging epistemic uncertainty estimation, our approach enables highly data-efficient active learning strategies.

Uncertainty quantification (UQ) is essential for reliable decision-making in safety-critical applications in probabilistic machine learning. For regression problems, dominant scalar UQ approaches - notably, those based on proper scoring rules - measure uncertainty via pointwise predictive risk. This can lead to counterintuitive results when the target statistic is not the conditional expectation. We propose an alternative framework, in which uncertainty is characterised by the volume of the most probable subset of a distribution's support. QUEST (Quantifying Uncertainty via highest dEnSiTy regions) is a novel approach to UQ based on the concentration of Lebesgue measure at a distribution's peak(s), evaluated at one or more values of a robustness parameter $α$. We establish connections between our measures and classical statistics from information theory and economics. We show that, unlike popular alternatives based on proper scoring rules, QUEST measures of epistemic and aleatoric uncertainty satisfy a set of axioms adapted from the UQ literature, including monotonicity under distributional spread and invariance to location shifts. Selective prediction benchmarks confirm that QUEST performs favourably against standard measures such as variance and differential entropy.

Conformal Prediction (CP) provides robust uncertainty guarantees for predictive models, but is typically applied post hoc, which misaligns model training with the conformal goal of producing efficient (i.e., narrow) intervals. We propose SPACR (Single-Pass Adaptive Conformal Regressor), a novel method for directly training uncertainty-aware regressors within a differentiable loss. SPACR jointly optimizes accuracy, efficiency, and validity without batch-splitting or a predefined confidence level during training. As a result, a single SPACR model yields valid prediction intervals at multiple confidence levels during inference, avoiding the costly retraining required by methods like Directly Optimized Inductive Conformal Regression (DOICR). Experiments on diverse tabular and image datasets show that SPACR consistently gives tighter intervals and better coverage-efficiency trade-offs compared to standard CP and DOICR, while significantly reducing computational costs relative to retraining-dependent baselines.

Symbolic regression (SR) is a class of methods that systematically explore the space of mathematical functions to discover models that accurately capture the underlying relationships in a dataset. Despite recent advances in the field, a lack of support for uncertainty quantification (UQ) limits its adoption in real-world decision processes. In regression analysis, UQ provides important information about the model reliability, which can both help to avoid overfitting by accounting for uncertainty in the data, and provide insights for decision-making. This survey is the first to clearly address this issue, with the objective of introducing essential UQ concepts and reviewing the current literature on UQ in SR, which can be broadly organized into three research directions: frequentist, Bayesian, and model selection. Despite its importance, UQ in SR is still underexplored, which motivates further research into reliable UQ methods for SR.

Providing non-conservative uncertainty quantification for function estimates derived from noisy observations remains a fundamental challenge in statistical machine learning, particularly for applications in safety-critical domains. In this work, we propose novel non-asymptotic probabilistic uniform error bounds for kernel-based regression. Compared to related bounds in the literature that are restricted to (conditionally) independent sub-Gaussian noise, our bounds allow to consider a broad class of non-Gaussian distributions, such as sub-Gaussian, bounded, sub-exponential, and variance/moment-bounded noise. Moreover, our results apply to correlated and uncorrelated noise. We compare our proposed error bounds with existing results in terms of the induced uncertainty region and their performance in safe control, demonstrating the tightness of the proposed bounds.

Edge learning refers to training machine learning models deployed on edge platforms, typically using new data accumulated onboard. The computational limitations on edge devices affect not only model optimisation, but also calculation of the predictive uncertainty of the current model on the unlabelled data, which is vital for informing model updating. In this paper, we investigate edge learning in the context of performing deep image regression on a remote sensing satellite, where a deep network is executed by an onboard computer to regress a scalar $y$ from an input image, e.g., $y$ is the percentage of pixels indicating cloud coverage or land use. We propose an uncertainty-guided edge learning (UGEL) algorithm that can accurately prioritise the data to speed up training convergence of the on-board regression model. Underpinning UGEL is the calculation of predictive uncertainty based on deep beta regression, where a deep network is used to estimate the parameters of a beta distribution for which the target $y$ for an input image has a high likelihood. Compared to established methods for uncertainty estimation that are either too costly on edge devices (e.g., require many forward passes per sample) or make strict assumptions on the predictive distribution (e.g., Gaussian), deep beta regression is computable in a single forward pass and allows more general predictive distributions. Results show that UGEL delivers faster-converging edge learning than active or semi-supervised learning. Code and models are publicly available at https://github.com/anh-vunguyen/UGEL.

Conformal Prediction provides distribution-free prediction intervals with guaranteed coverage, but its reliance on a single global calibration threshold obscures the sources of uncertainty at the instance level. In particular, it conflates irreducible noise with uncertainty induced by heterogeneous training data (aleatoric), model limitations, or calibration mismatch (epistemic), offering little insight into why an interval is wide or whether it could be reduced. We introduce an uncertainty-aware explainability framework that analyses the reducibility of calibration-induced epistemic conformal uncertainty via progressive calibration localisation for regression tasks. The approach is diagnostic rather than causal: it does not estimate true aleatoric or epistemic uncertainty, but explains how conformal intervals contract and stabilise as calibration support is localised around a test instance. Across benchmarks and real-world data, absolute reducible uncertainty aligns with epistemic proxies, while its relative contribution varies by task, revealing regimes hidden by interval width. This instance-level view complements conformal uncertainty, enhancing interpretability without altering the predictor or coverage.

Neural activation coverage (NAC) is a recently-proposed technique for out-of-distribution detection and generalization. We build upon this promising foundation and extend the method to work as an uncertainty estimation technique for already-trained artificial neural networks in the domain of regression. Our experiments confirm NAC uncertainty scores to be more meaningful than other techniques, e.g. Monte-Carlo Dropout.

Conformal prediction delivers prediction intervals with distribution-free coverage, but its intervals can look overconfident in regions where the model is extrapolating, because standard conformal scores do not explicitly represent epistemic uncertainty. Credal methods, by contrast, make epistemic effects visible by working with sets of plausible predictive distributions, but they are typically model-based and lack calibration guarantees. We introduce CREDO, a simple "credal-then-conformalize" recipe that combines both strengths. CREDO first builds an interpretable credal envelope that widens when local evidence is weak, then applies split conformal calibration on top of this envelope to guarantee marginal coverage without further assumptions. This separation of roles yields prediction intervals that are interpretable: their width can be decomposed into aleatoric noise, epistemic inflation, and a distribution-free calibration slack. We provide a fast implementation based on trimming extreme posterior predictive endpoints, prove validity, and show on benchmark regressions that CREDO maintains target coverage while improving sparsity adaptivity at competitive efficiency.

Uncertainty quantification (UQ) in deep learning regression is of wide interest, as it supports critical applications including sequential decision making and risk-sensitive tasks. In heteroskedastic regression, where the uncertainty of the target depends on the input, a common approach is to train a neural network that parameterizes the mean and the variance of the predictive distribution. Still, training deep heteroskedastic regression models poses practical challenges in the trade-off between uncertainty quantification and mean prediction, such as optimization difficulties, representation collapse, and variance overfitting. In this work we identify previously undiscussed fallacies and propose a simple and efficient procedure that addresses these challenges jointly by post-hoc fitting a variance model across the intermediate layers of a pretrained network on a hold-out dataset. We demonstrate that our method achieves on-par or state-of-the-art uncertainty quantification on several molecular graph datasets, without compromising mean prediction accuracy and remaining cheap to use at prediction time.

We propose a robust Bayesian formulation of random feature (RF) regression that accounts explicitly for prior and likelihood misspecification via Huber-style contamination sets. Starting from the classical equivalence between ridge-regularized RF training and Bayesian inference with Gaussian priors and likelihoods, we replace the single prior and likelihood with $ε$- and $η$-contaminated credal sets, respectively, and perform inference using pessimistic generalized Bayesian updating. We derive explicit and tractable bounds for the resulting lower and upper posterior predictive densities. These bounds show that, when contamination is moderate, prior and likelihood ambiguity effectively acts as a direct contamination of the posterior predictive distribution, yielding uncertainty envelopes around the classical Gaussian predictive. We introduce an Imprecise Highest Density Region (IHDR) for robust predictive uncertainty quantification and show that it admits an efficient approximation via an adjusted Gaussian credible interval. We further obtain predictive variance bounds (under a mild truncation approximation for the upper bound) and prove that they preserve the leading-order proportional-growth asymptotics known for RF models. Together, these results establish a robustness theory for Bayesian random features: predictive uncertainty remains computationally tractable, inherits the classical double-descent phase structure, and is improved by explicit worst-case guarantees under bounded prior and likelihood misspecification.

Conformal regression provides finite-sample marginal coverage, but interval efficiency depends on how uncertainty is allocated across inputs. We propose Conformal Laplace-Aware Predictive Scaling (CLAPS), which combines a learned heteroscedastic scale with last-layer Laplace uncertainty. The learned variance also weights the training features used to construct the posterior, yielding a noise-weighted support correction in the learned representation. The correction contracts as weighted feature information increases and becomes most influential under localized support imbalance. Experiments show near-nominal marginal coverage and modest efficiency improvements over the matched aleatoric baseline on real regression benchmarks.

Regression tasks, notably in safety-critical domains, require reliable uncertainty quantification, yet the literature remains largely classification-focused. To address this, we introduce a family of measures for total, aleatoric, and epistemic uncertainty in multivariate regression based on strictly proper kernel scores. The framework provides a principled recipe for designing new uncertainty measures whose behavior, such as tail sensitivity or out-of-distribution responsiveness, is governed by the choice of the underlying kernel, while also encompassing existing measures under a joint analysis. We prove explicit correspondences between properties of the kernel and behavior of resulting uncertainty measures, yielding concrete design guidelines for practitioners. Extensive experiments across structured regression tasks, including spatial and functional domains, demonstrate effectiveness on downstream tasks such as out-of-distribution detection and active learning, and reveal that different kernel choices lead to distinct trade-offs, offering practitioners guidance for task-specific selection.

This paper introduces a framework for uncertainty quantification in regression models defined on metric spaces. Using a proposed notion of homoscedasticity, we define a conformal prediction algorithm that provides finite-sample marginal coverage guarantees and fast convergence rates to the oracle prediction region. For heteroscedastic settings, we introduce a kNN procedure that yields locally adaptive prediction radii in general metric spaces. Although this procedure does not provide the same finite-sample guarantees as the conformal algorithm, it is designed to improve local coverage calibration without imposing smoothing assumptions. Both procedures are compatible with a broad range of regression algorithms and scale to large datasets, allowing practitioners to use their preferred models and incorporate domain-specific knowledge. Building on the heteroscedastic $k$NN approach, we also develop a flexible sequential extension for metric-space-valued time series based on nearest-neighbor expert aggregation. We establish the consistency of the proposed estimators under minimal conditions. Finally, we illustrate the practical utility of our framework in personalized medicine applications involving random objects such as probability distributions and graph Laplacians.

Accurate uncertainty quantification is critical for reliable predictive modeling. Existing methods typically address either aleatoric uncertainty due to measurement noise or epistemic uncertainty resulting from limited data, but not both in a balanced manner. We propose CLEAR, a calibration method with two distinct parameters, $γ_1$ and $γ_2$, to combine the two uncertainty components and improve the conditional coverage of predictive intervals for regression tasks. CLEAR is compatible with any pair of aleatoric and epistemic estimators; we show how it can be used with (i) quantile regression for aleatoric uncertainty and (ii) ensembles drawn from the Predictability-Computability-Stability (PCS) framework for epistemic uncertainty. Across 17 diverse real-world datasets, CLEAR achieves an average improvement of 28.3% and 17.5% in the interval width compared to the two individually calibrated baselines while maintaining nominal coverage. Similar improvements are observed when applying CLEAR to Deep Ensembles (epistemic) and Simultaneous Quantile Regression (aleatoric). The benefits are especially evident in scenarios dominated by high aleatoric or epistemic uncertainty. Project page: https://unco3892.github.io/clear/

Uncertainty quantification is crucial in machine learning, yet most (axiomatic) studies of uncertainty measures focus on classification, leaving a gap in regression settings with limited formal justification and evaluations. In this work, we provide a formal way of representing uncertainty in continuous space, using a general parametric formulation, allowing for tractable analysis and evaluation of uncertainty measures. Within this framework, we propose a set of axioms that enable rigorous assessment of total, aleatoric, and epistemic uncertainty measures. Together, this allows for a theoretical examination of uncertainty measures and their corresponding properties. As a specific example, we compare the widely used entropy- and variance-based measures with respect to established predictive models and analyze their limitations and challenges in uncertainty quantification. Our work provides a principled way to understand and develop uncertainty measures in supervised regression, offering theoretical insights and practical guidelines for reliable uncertainty assessment.